Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

Abstract

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations ${\mathcal{I}}_{\lambda }^n$ of the rank $⩽n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.

With the participation of the new coauthor - Jimmie Lawson - the manuscript has been substantially revised and expanded. Accordingly, we have also changed the manuscript title

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